Learn Russian: Дети обычно приоткрывают окно ночью. - The children usually crack the window open at night.

The prefix при- is used here to indicate an incomplete, slight, or gentle action (the attenuative meaning). While открывают means 'they open', приоткрывают means 'they open slightly' or 'they crack open'. This is a very common use of при- with verbs of physical action.

Since cracking a window is a completed physical result, why isn't the perfective verb приоткроют used?

The adverb обычно (usually) tells us this is a repeated, habitual action. In Russian, habitual actions almost always require the imperfective aspect, even if each individual instance has a clear result. Therefore, we use the imperfective приоткрывают instead of the perfective приоткроют.

Does adding the attenuative prefix при- change the case of the object the verb takes?

No. Just like the base verb открывать (to open) takes a direct object in the accusative case, the prefixed verb приоткрывать also takes the accusative (окно). Prefixes that just modify the degree of the action usually do not alter the verb's underlying grammatical rules.

Why is 'at night' translated as ночью without any preposition?